\(\frac{(39,897)(0.0096)}{198.76}\) is approximately

1. Translate the problem requirements: We need to calculate the value of a fraction where the numerator is the product of approximately 40,000 and 0.01, divided by approximately 200. The answer choices differ by factors of 10, so approximation will be sufficient.
2. Round numbers to friendly values: Convert each number to a simpler form that preserves the order of magnitude but makes mental math easier.
3. Apply fraction simplification logic: Rearrange the calculation to group similar orders of magnitude and identify what cancels out.
4. Estimate the final result: Perform the simplified calculation and compare with answer choices to identify the correct order of magnitude.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're being asked to do here. We have a fraction with a big number times a small decimal on top, divided by another number on the bottom. The expression is:

\(\frac{(39,897)(0.0096)}{198.76}\)

When I look at the answer choices, I notice they range from 0.02 all the way up to 200. That's a huge range! Each answer choice is about 10 times bigger than the previous one. This tells me that I don't need to calculate this precisely - I just need to get in the right ballpark.

Process Skill: TRANSLATE - Converting the mathematical expression into an estimation problem based on answer choice patterns

2. Round numbers to friendly values

Now I'm going to make this calculation much easier by rounding each number to something I can work with mentally.

Let me look at each number:

• 39,897 is very close to 40,000 (I'm rounding to the nearest ten thousand)
• 0.0096 is very close to 0.01 (I'm rounding to the nearest hundredth)
• 198.76 is very close to 200 (I'm rounding to the nearest hundred)

So my expression becomes:

\(\frac{(40,000)(0.01)}{200}\)

This is so much friendlier to work with!

3. Apply fraction simplification logic

Now I'll work through this step by step, doing the easiest operations first.

First, let me calculate the numerator:
\((40,000)(0.01)\)

When I multiply by 0.01, I'm essentially moving the decimal point two places to the left:
\(40,000 \times 0.01 = 400\)

So now my expression is:

\(\frac{400}{200}\)

This is a simple division: \(400 \div 200 = 2\)

Process Skill: SIMPLIFY - Breaking down the calculation into manageable steps and using decimal place logic

4. Estimate the final result

My calculation gives me approximately 2.

Let me check this against the answer choices:

1. 0.02 (much smaller than my result)
2. 0.2 (still smaller than my result)
3. 2 (matches my result perfectly!)
4. 20 (10 times bigger than my result)
5. 200 (100 times bigger than my result)

The answer is C. 2.

To double-check my reasoning: I had a large number (about 40,000) times a small decimal (about 0.01), which gave me 400, then divided by 200, which gave me 2. This makes sense!

Common Faltering Points

Errors while devising the approach

1. Attempting exact calculation instead of estimation: Students often miss that this is an approximation problem and try to calculate the exact value of (39,897)(0.0096)/198.76, leading to unnecessarily complex arithmetic that's prone to errors.

2. Poor rounding strategy: Students may round numbers in ways that make mental math difficult (like rounding 39,897 to 39,900 instead of 40,000) or round too conservatively, missing the opportunity to simplify dramatically.

3. Ignoring answer choice patterns: Students fail to notice that answer choices differ by factors of 10 (0.02, 0.2, 2, 20, 200), which signals that rough estimation rather than precise calculation is sufficient.

Errors while executing the approach

1. Decimal multiplication errors: When calculating 40,000 × 0.01, students commonly make decimal point errors, such as getting 4,000 instead of 400, by incorrectly moving the decimal point.

2. Order of operations confusion: Students might incorrectly calculate the expression by doing division before multiplication, computing 39,897/(198.76 × 0.0096) instead of (39,897 × 0.0096)/198.76.

3. Mental arithmetic mistakes: Even with simplified numbers like 400/200, students may rush and make basic division errors, especially under time pressure.

Errors while selecting the answer

1. Decimal place confusion: After correctly calculating 400/200 = 2, students might second-guess themselves and select 0.2 or 20, thinking they made a decimal error in their estimation process.

2. Over-correction for rounding: Students may incorrectly assume their rounded estimates are too far off and choose a different answer choice, not trusting that their systematic rounding approach was valid.